Enumeration exercises
• Students and classes (easy)
– There is a makeup midterm for two students at room 280 McDonald Eng. Building. There are 64
places to sit. In how many ways can the students sit down and take the exam? (they can sit next to
each other)
64 × 63
– There were 85 students taking the midterm. As they hand in the exam the prof. would put the
booklets in a pile. How many possible orderings of the booklets are there?
85!
– There are 24 students in a math contest. How many possible outcomes are there for the first, second
and tirhd prizes are there? (No ties are possible)
24 × 23 × 22
• Poker questions
– How distinct many pairs are there in a deck of cards?
13 × C(4, 2) = 13 × 6
– How distinct many ’three of a kind’ are there in a deck of cards?
13 × C(4, 3) = 13 × 4
– How many distinct ’full-house’ are there in a deck of cards?
13 × C(4, 2) × 12 × C(4, 3) = 13 × 6 × 12 × 4
– How many ’straight flush’ are there in a deck of cards?
4 × 10
• Car plates
– If car plates consists of 3 letters and 3 digits. What is the number of distinct possible car plates?
263 × 103
– If car plates consists of 3 distinct letters and 3 distinct digits. What is the number of distinct possible
car plates?
(26 × 25 × 24) × (10 × 9 × 8)
– If car plates consists of 3 letters and 3 digits. And there are no plates with 3 equal digits nor 3 equal
letters. What is the number of distinct possible car plates?
(263 − 26) × (103 − 10)
• Passwords and bit-strings
– The passwords on a computer system are 6 to 8 bits long and are formed by digits or uppercase letters.
How many distinct passwords are there?
366 + 367 + 368
1
– The passwords on a computer system are 6 bits long, are formed by digits or uppercase letters and at
least one digit. How many distinct passwords are there?
P6
366 − 266 = 10 i=1 26i−1 366−i
– How many bit strings of length 8 either start with a 1 bit or end with the two bits 00?
27 + 26 − 25 = 25 (4 + 1) = 25 (4 + 2 − 1)
• Functions ans sets
– Let A and B be finite sets of cardinality a and b respectively. How many functions from A to B are
there?
ba
– Let A and B be finite sets of cardinality a and b respectively. If b ≥ a, how many injective functions
from A to B are there?
P (b, a) = b × (b − 1) × ... × (b − a + 1)
– Let A and B be finite sets of cardinality a and b respectively. If a > b, how many one-to-one functions
from A to B are there?
0
– If a set A has cardinality k. What is the cardinality of the power set of A?
2k
• Students and classes (hard)
– There are 100 students in a math contest. There will be 10 gold medals 15 silver medals and 30 bronze
medals. In how many ways can these prizes be delivered?
C(100, 10) × C(90, 15) × C(75, 30) = C(100, 55) × C(55, 10) × C(45, 15)
– There is a makeup midterm for two students at room 280 McDonald Eng. Building. The student will
sit in the first row, where there are 8 places, and can not sit together. In how many ways can the
students sit down and take the exam?
2 × (6 + 5 + 4 + 3 + 2 + 1) = (2 × 5) + (5 × 6) = 42
– There are 24 students in a class of robotics. They will split the work in 4 teams of 6; each team in
charge of a distinct task. In how many ways can the teams be formed?
C(24, 6) × C(18, 6) × C(12, 6) =
24!
(6!)4
– There are 24 students will make projects, in teams of 6 people, for a class. In how many ways can the
teams be formed?
[C(24, 6) × C(18, 6) × C(12, 6)]/4! =
24!
4!(6!)4
2